Reionization process dependence of the ratio of CMB polarization power spectra at low-$\ell$

We investigate how much the ratio of cosmic microwave background (CMB) polarization power spectra $C^{BB}_\ell/C^{EE}_\ell$ at low-$\ell$ ($\ell \lesssim 10$) depends on the process of reionization. Both such low-$\ell$ B-mode and E-mode polarization powers are dominantly produced by Thomson scattering of CMB photons off the free electrons which are produced in the process of reionization. Since the reionization should be finished until at least the redshift $z \simeq 6$ and the low-$\ell$ polarization powers are produced at late time, the ratio is rather insensitive by the ionization process at higher redshifts, but it is sensitive to the value of optical depth.The value of the ratio at $\ell=2$, however, is almost insensitive to the reionization process including the value of optical depth, and the value is approximately half of the value of tensor-to-scalar ratio. This fact can be utilized for future determination of tensor-to-scalar ratio in spite of the ambiguity due to cosmic variance.